Question

Factor completely.
3, x, cubed, minus, 5, x, squared, minus, 27, x, plus, 45
3x^3−5x^2−27x+45

Answer 1

To factor completely, we can first use the Rational Root Theorem to find any possible rational roots.

The Rational Root Theorem states that if a polynomial:

p(x) = a_n * x^n + a_n-1 * x^(n-1) + ... + a_1 * x + a_0

has a rational root x = p/q, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n.

In our case, the constant term is 45, and the leading coefficient is 3.

The factors of 45 are ±1, ±3, ±5, ±9, ±15, ±45.
The factors of 3 are ±1, ±3.

Using the Rational Root Theorem, the possible rational roots are:
±1, ±3, ±5, ±9, ±15, ±45(possible numerators) /
±1, ±3(possible denominators)

Let's try some of these possibilities by substituting them into the polynomial:

For x = 1:
3(1)^3 - 5(1)^2 - 27(1) + 45 = 3 - 5 - 27 + 45 = 16, which is not equal to 0.

For x = -1:
3(-1)^3 - 5(-1)^2 - 27(-1) + 45 = -3 - 5 + 27 + 45 = 64, which is not equal to 0.

For x = 3:
3(3)^3 - 5(3)^2 - 27(3) + 45 = 81 - 45 - 81 + 45 = 0.

Therefore, x = 3 is a root.

Using synthetic division, we can divide the polynomial by x - 3:

```
3 | 3 -5 -27 45
| 9 12 -45
+____________________
3 4 -15 0
```

The quotient is 3x^2 + 4x - 15, and the remainder is 0.

So, we have factored the polynomial as (x - 3)(3x^2 + 4x - 15).

Next, we can factor the quadratic expression 3x^2 + 4x - 15.

We look for two numbers whose product is -45 (the product of the coefficient of x^2 and the constant term), and whose sum is 4 (the coefficient of x).
The numbers that fit these conditions are 9 and -5.

So, we can factor 3x^2 + 4x - 15 as (x + 3)(3x - 5).

Therefore, the completely factored form of the polynomial 3x^3 - 5x^2 - 27x + 45 is:
(x - 3)(x + 3)(3x - 5).